10.8.2 Vibration Configuration Interaction Theory

To solve the nuclear vibrational Schrödinger equation, one can only use direct integration procedures for diatomic molecules. For larger systems, a truncated version of full configuration interaction is considered to be the most accurate approach. When one applies the variational principle to the vibrational problem, a basis function for the nuclear wave function of the $n$th excited state of mode $i$ is

where the ${\varphi }_i^{(n)}$ represents the harmonic oscillator eigenfunctions for normal mode $q_i$. This can be expressed in terms of Hermite polynomials:

With the basis function defined in Eq. (10.66), the $n$th wave function can be described as a linear combination of the Hermite polynomials:

where $n_i$ is the number of quanta in the $i$th mode. We propose the notation VCI($n$) where $n$ is the total number of quanta, i.e.:

To determine this expansion coefficient $c^{(n)}$, we integrate the $\widehat{H}$, as in Eq. (4.1), with ${\mathrm{\Psi }}^{(n)}$ to get the eigenvalues

This gives us frequencies that are corrected for anharmonicity to $n$ quanta accuracy for a $m$-mode molecule. The size of the secular matrix on the right hand of Eq. (10.70) is ${((n+m)!/n!m!)}^2$, and the storage of this matrix can easily surpass the memory limit of a computer. Although this method is highly accurate, we need to seek for other approximations for computing large molecules.